Calculating Frictional Force During Pure Rolling

Pure rolling is a type of motion where an object rolls without slipping. This means that the point of the object in contact with the surface is momentarily at rest with respect to the surface. When an object is purely rolling, the frictional force acts to prevent slipping and is often referred to as static friction.

Understanding Pure Rolling

Before we delve into the calculation of frictional force during pure rolling, it's important to understand the conditions for pure rolling. For an object to roll purely, the following condition must be met:

[ v_{CM} = r \omega ]

Where:

• ( v_{CM} ) is the velocity of the center of mass of the rolling object.
• ( r ) is the radius of the rolling object.
• ( \omega ) is the angular velocity of the rolling object.

Frictional Force in Pure Rolling

In pure rolling, the frictional force does not do any work since there is no relative motion between the point of contact and the surface. The frictional force is static in nature and its maximum value is given by:

[ f_{max} = \mu_s N ]

Where:

• ( f_{max} ) is the maximum static frictional force.
• ( \mu_s ) is the coefficient of static friction.
• ( N ) is the normal force acting on the rolling object.

However, the actual frictional force during pure rolling can be less than or equal to ( f_{max} ) and is determined by the external forces and torques acting on the object.

Calculating Frictional Force

To calculate the frictional force during pure rolling, we need to consider the forces and torques acting on the object. The frictional force can be calculated using the following steps:

1. Identify all the forces acting on the object.
2. Apply Newton's second law to the translational motion of the object's center of mass.
3. Apply Newton's second law to the rotational motion about the center of mass.
4. Use the condition for pure rolling to relate translational and rotational motion.
5. Solve the equations to find the frictional force.

Example Calculation

Let's consider a solid sphere of mass ( m ) and radius ( r ) rolling down an inclined plane with an angle ( \theta ) to the horizontal.

1. Forces Acting on the Sphere:

   • Gravitational force ( mg ) acting downward.
   • Normal force ( N ) acting perpendicular to the inclined plane.
   • Frictional force ( f ) acting parallel to the inclined plane and opposite to the direction of slipping.

2. Translational Motion: [ mg \sin(\theta) - f = m a_{CM} ] Where ( a_{CM} ) is the acceleration of the center of mass.

3. Rotational Motion: [ f r = I \alpha ] Where ( I ) is the moment of inertia of the sphere and ( \alpha ) is the angular acceleration.

4. Moment of Inertia for a Solid Sphere: [ I = \frac{2}{5} m r^2 ]

5. Condition for Pure Rolling: [ a_{CM} = r \alpha ]

6. Solving the Equations: Substituting ( I ) and the pure rolling condition into the rotational motion equation, we get: [ f r = \frac{2}{5} m r^2 \left( \frac{a_{CM}}{r} \right) ] Simplifying, we find: [ f = \frac{2}{5} m a_{CM} ]

Substituting ( f ) into the translational motion equation, we get: [ mg \sin(\theta) - \frac{2}{5} m a_{CM} = m a_{CM} ] Solving for ( a_{CM} ), we find: [ a_{CM} = \frac{5}{7} g \sin(\theta) ]

Finally, substituting ( a_{CM} ) back into the expression for ( f ), we get: [ f = \frac{2}{5} m \left( \frac{5}{7} g \sin(\theta) \right) ] [ f = \frac{2}{7} m g \sin(\theta) ]

Summary Table

Aspect	Pure Rolling Condition
Velocity Relation	( v_{CM} = r \omega )
Frictional Force Nature	Static
Work Done by Friction	Zero
Frictional Force Formula	( f \leq \mu_s N )
Example Object	Solid Sphere
Moment of Inertia (I)	( \frac{2}{5} m r^2 )

In conclusion, calculating the frictional force during pure rolling requires an understanding of both translational and rotational dynamics, as well as the condition that relates the two. By applying Newton's second law to both types of motion and using the pure rolling condition, one can determine the frictional force necessary to maintain pure rolling without slipping.